Related papers: On Parameter Differentiation for Integral Represen…
Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and…
Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial…
In this paper we study the Hilbert scales defined by the associated Legendre functions for arbitrary integer values of the parameter. This problem is equivalent to study the left-definite spectral theory associated to the modified Legendre…
We derive explicit expressions for the parameter derivatives $[\partial^{2}P_{\nu}(z)/\partial\nu^{2}]_{\nu=0}$ and $[\partial^{3}P_{\nu}(z)/\partial\nu^{3}]_{\nu=0}$, where $P_{\nu}(z)$ is the Legendre function of the first kind. It is…
We express a certain complex-valued solution of Legendre's differential equation as the product of an oscillatory exponential function and an integral involving only nonoscillatory elementary functions. By calculating the logarithmic…
We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as…
This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of…
One may consider the generalization of Jacobi polynomials and the Jacobi function of the second kind to a general function where the index is allowed to be a complex number instead of a non-negative integer. These functions are referred to…
Series involving hypergeometric functions are used to derive, extend and evaluate integrals involving the product of two Bessel functions of the first kind $J_{u}(a z)$ $J_{v}(b z)$ with order $u,v$, studied by Landau et al. The method used…
An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this,…
Expressions for the derivatives of the Legendre polynomials of the first kind with respect to the order of these polynomials are given. An explicit form for the fourth derivative is presented.
A Mellin transform representation is derived for the associated Legendre functions of the first kind on the cut
The main objective of this paper is to give a wide study on the conformable fractional Legendre polynomials (CFLPs). This study is assumed to be a generalization and refinement, in an easy way, of the scalar case into the context of the…
A new integral identity for functions with continuous second partial derivatives is derived. It is shown that the value of any function f(r,t) at position r and time t is completely determined by its previous values at all other locations…
We improve bounds on the degree and sparsity of Boolean functions representing the Legendre symbol as well as on the $N$th linear complexity of the Legendre sequence. We also prove similar results for both the Liouville function for…
We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function ${}_p\Psi_q(z)$ of one variable. Some consequences of…
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
In this paper we study fractional powers of the Bessel differential operator defined on a semiaxis. Some important properties of such fractional powers of the Bessel differential operator are proved. They include connections with Legendre…
Spherical Bessel functions appear commonly in many areas of physics wherein there is both translation and rotation invariance, and often integrals over products of several arise. Thus, analytic evaluation of such integrals with different…
We develop a compositional approach for automatic and symbolic differentiation based on categorical constructions in functional analysis where derivatives are linear functions on abstract vectors rather than being limited to scalars,…